Biharmonic functions on Schrodinger networks

In an infinite graph X, the equation Delta u(x) = phi(x)u(x), phi(x) >= 0, is termed the Schrodinger equation. The solutions of this equation are named phi.-harmonic functions. This article studies varied aspects associated with these solutions: Harnack principle, minimum principle, domination principle, Dirichlet solution etc. The main thrust is the development of an abstract theory of discrete phi-biharmonic functions on X and use it for the phi-biharmonic classification of infinite graphs.